Let be the abelian variety attached by Shimura to a normalized newform and assume that has elliptic quotients. The paper deals with the determination of the one dimensional subspaces (elliptic directions) in corresponding to the pullbacks of the regular differentials of all elliptic quotients of . For modular elliptic curves over number fields without complex multiplication (CM), the directions were studied by the authors in . The main goal of the present paper is to characterize the directions corresponding to elliptic curves with CM. Then we apply the results obtained to the case , for primes and mod . For this case we prove that if has CM, then all optimal elliptic quotients of are also optimal in the sense that its endomorphism ring is the maximal order of . Moreover, if has trivial Nebentypus then all optimal quotients are Gross’s elliptic curve and its Galois conjugates. Among all modular parametrizations , we describe a canonical one and discuss some of its properties.
Cite this article
Josep González Rovira, Joan-Carles Lario, Modular elliptic directions with complex multiplication (with an application to Gross’s elliptic curves). Comment. Math. Helv. 86 (2011), no. 2, pp. 317–351DOI 10.4171/CMH/225